The equations of fluid motion

Transcription

1 Chapter 6 The equations of fluid motion In order to proceed further with our discussion of the circulation of the atmosphere, and later the ocean, we must develop some of the underlying theory governing the motion of a fluid on the spinning Earth. A differentially heated, stratified fluid on a rotating planet cannot move in arbitrary paths. Indeed, there are strong constraints on its motion imparted by the angular momentum of the spinning Earth. These constraints are profoundly important in shaping the pattern of atmosphere and ocean circulation and their ability to transport properties around the globe. The laws governing the evolution of both fluids are the same and so our theoretical discussion willnotbespecifictoeitheratmosphereorocean,butcanandwillbeapplied to both. Because the properties of rotating fluids are often counter-intuitive and sometimes difficult to grasp, alongside our theoretical development we will describe and carry out laboratory experiments with a tank of water on a rotating table (Fig.6.1). Many of the laboratory experiments we make use of are simplified versions of classics that have become cornerstones of geophysical fluid dynamics. They are listed in Appendix Furthermore we have chosen relatively simple experiments that, in the main, do nor require sophisticated apparatus. We encourage you to have a go or view the attendant movie loops that record the experiments carried out in preparation of our text. We now begin a more formal development of the equations that govern the evolution of a fluid. A brief summary of the associated mathematical concepts, definitions and notation we employ can be found in an Appendix

2 154 CHAPTER 6. THE EQUATIONS OF FLUID MOTION Figure 6.1: Throughout our text, running in parallel with a theoretical development of the subject, we study the constraints on a differentially heated, stratified fluid on a rotating planet (left), by making use of laboratory analogues designed to illustrate the fundamental processes at work (right). A complete list of the laboratory experiments can be found in Section Differentiation following the motion When we apply the laws of motion and thermodynamics to a fluid to derive the equations that govern its motion, we must remember that these laws apply to material elements of fluid which are usually mobile. We must learn, therefore, how to express the rate of change of a property of a fluid element, following that element as it moves along,ratherthanatafixed point in space. It is useful to consider the following simple example. Consider again the situation sketched in Fig.4.13 in which a wind blows over a hill. The hill produces a pattern of waves in its lee. If the air is sufficiently saturated in water vapor, the vapor often condenses out to form cloud at the ridges of the waves as described in Section 4.4 and seen in Figs.4.14 and Let us suppose that a steady state is set up so the pattern of cloud does not change in time. If C = C(x, y, z, t) is the cloud amount, where (x, y) are horizontal coordinates, z is the vertical coordinate, t is time, then:

3 6.1. DIFFERENTIATION FOLLOWING THE MOTION 155 µ C =0, t fixed point in space wherewekeepatafixed point in space, but at which, because the air is moving, there are constantly changing fluid parcels. The derivative t fixed point is called the Eulerian derivative after Euler 1. But C is not constant following along a particular parcel; as the parcel movesupwardsintotheridgesofthewave,itcools,watercondensesout, cloud forms, and so C increases (recall GFD Lab 1, Section 1.3.3); as the parcel moves down into the troughs it warms, the water goes back in to the gaseous phase, the cloud disappears and C decreases. Thus µ C t fixed particle even though the wave-pattern is fixed in space and constant in time. So, how do we mathematically express differentiation following the motion? In order to follow particles in a continuum a special type of differentiation is required. Arbitrarily small variations of C(x, y, z, t), a function of position and time, are given to the first order by: δc = C t δt + C x 6= 0 C C δx + δy + y z δz where the partial derivatives etc. are understood to imply that the other t variables are kept fixed during the differentiation. The fluid velocity is the 1 Leonhard Euler ( ). Euler made vast contributions to mathematics in the areas of analytic geometry, trigonometry, calculus and number theory. He also studied continuum mechanics, lunar theory, elasticity, acoustics, the wave theory of light, hydraulics and laid the foundation of analytical mechanics. In the 1750 s Euler published a number of major pieces of work setting up the main formulas of fluid mechanics, the continuity equation and the Euler equations for the motion of an inviscid incompressible fluid.

4 156 CHAPTER 6. THE EQUATIONS OF FLUID MOTION rate of change of position of the fluid element, following that element along. The variation of a property C following an element of fluid is thus derived by setting δx = uδt, δy = vδt, δz = wδt, whereu is the speed in the x-direction, v is the speed in the y-direction and w is the speed in the z-direction, thus: (δc) fixed particle = µ C t + u C x + v C y + w C δt z where (u, v, w) is the velocity of the material element which by definition is the fluid velocity. Dividing by δt andinthelimitofsmallvariations wesee that: µ C t fixed particle = C t + u C x + v C y + w C z = DC inwhichweusethesymbol D toidentifytherateofchangefollowingthe motion: D t + u x + v y + w z + u.. (6.1) t ³ Here u =(u, v, w) is the velocity vector and,, is the gradient x y z D operator. is called the Lagrangian derivative (after Lagrange; ) [it is also called variously the substantial, the total or the material derivative]. Itsphysicalmeaningis timerateofchangeofsomecharacteristic of a particular element of fluid (which in general is changing its position). By contrast, as introduced above, the Eulerian derivative, expresses the rate t of change of some characteristic at a fixed point in space (but with constantly changing fluid element because the fluid is moving). d Some writers use the symbol for the Lagrangian derivative, but this dt is better reserved for the ordinary derivative of a function of one variable, the sense it is usually used in mathematics. Thus, for example, the rate of change of the radius of a rain drop would be written dr with the identity dt of the drop understood to be fixed. Inthesamecontext D could refer to the motion of individual particles of water circulating within the drop itself. Another example is the vertical velocity, defined as w = Dz/: ifonesits in an air parcel and follow it around, w istherateatwhichone sheight changes 2. 2 Meteorologists like working in pressure coordinates in which p is used as a vertical

5 6.2. EQUATION OF MOTION FOR A NON-ROTATING FLUID 157 The term u. in Eq.(6.1) represents advection and is the mathematical representation of the ability of a fluid to carry its properties with it as it moves. For example, the effects of advection are evident to us every day. Inthenorthernhemispheresoutherlywinds(fromthesouth)tendtobe warm and moist because the air carries with it properties typical of tropical latitudes; northerly winds tend to be cold and dry because they advect properties typical of polar latitudes. We will now use the Lagrangian derivative to help us apply the laws of mechanics and thermodynamics to a fluid. 6.2 Equation of motion for a non-rotating fluid The state of the atmosphere or ocean at any time is defined by five key variables: u =(u, v, w); p and T, (six if we include specific humidity in the atmosphere, or salinity in the ocean). Note that by making use of the equation of state, Eq.(1.1), we can infer ρ from p and T. To tie these variables down we need five independent equations. They are: 1. the laws of motion applied to a fluid parcel; this yields three independent equations in each of the three orthogonal directions 2. conservation of mass 3. the law of thermodynamics, a statement of the thermodynamic state in which the motion takes place. These equations, five in all, together with appropriate boundary conditions, are sufficient to determine the evolution of the fluid. coordinate rather than z. In this coordinate an equivalent definition of vertical velocity is: ω = Dp, the rate at which pressure changes as the air parcel moves around. Since pressure varies much more quickly in the vertical than in the horizontal, this is still, for all practical purposes, a measure of vertical velocity, but expressed in units of h Pa s 1.Notealsothat upward motion has negative ω.

6 158 CHAPTER 6. THE EQUATIONS OF FLUID MOTION Figure 6.2: An elementary fluid parcel, conveniently chosen to be a cube of side δx, δy, δz, centered on (x, y, z). The parcel is moving with velocity u Forces on a fluid parcel We will now consider the forces on an elementary fluid parcel, of infinitesimal dimensions (δx, δy, δz) in the three coordinate directions, centered on (x, y, z) (see Fig.6.2). Since the mass of the parcel is δm = ρδxδyδz, then, when subjected to a net force F, Newton s Law of Motion for the parcel is ρδxδyδz Du = F, (6.2) where u is the parcel s velocity. As discussed earlier we must apply Eq.(6.2) to the same material mass of fluid, i.e., we must follow the same parcel around. Therefore, the time derivative in Eq.(6.2) is the total derivative, definedineq.(6.1),whichinthiscaseis Du = u t + u u x + v u y + w u z = u +(u ) u. t

7 6.2. EQUATION OF MOTION FOR A NON-ROTATING FLUID 159 Figure 6.3: Pressure gradient forces acting on the fluid parcel. The pressure of the surrounding fluid applies a force to the right on face A and to the left on face B. Gravity The effect of gravity acting on the parcel in Fig.6.2 is straightforward: the gravitational force is gδm, and is directed downward, F gravity = gρbz δx δy δz, (6.3) where bz is the unit vector in the upward direction and g is assumed constant. Pressure gradient Another kind of force acting on a fluid parcel is the pressure force within the fluid. Consider Fig.6.3. On each face of our parcel there is a force (directed inward) acting on the parcel equal to the pressure on that face multiplied by the area of the face. On face A, for example, the force is F (A) =p(x δx 2,y,z) δy δz, directed in the positive x-direction. Notethatwehaveusedthevalueofp at the mid-point of the face, which is valid for small δy, δz. OnfaceB, thereis

8 160 CHAPTER 6. THE EQUATIONS OF FLUID MOTION an x-directed force F (B) = p(x + δx 2,y,z) δy δz, which is negative (toward the left). Since these are the only pressure forces acting in the x-direction, the net x-component of the pressure force is δx2 δx2 F x = p(x,y,z) p(x +,y,z) δy δz. If we perform a Taylor expansion about the midpoint of the parcel, we have p(x + δx µ p 2,y,z) = p(x, y, z)+δx, 2 x p(x δx 2,y,z) = p(x, y, z) δx µ p, 2 x where the pressure gradient is evaluated at the midpoint of the parcel, and where we have neglected the small terms of O(δx 2 ) and higher. Therefore the x-component of the pressure force is F x = p δx δy δz. x It is straightforward to apply the same procedure to the faces perpendicular to the y-andz-directions, to show that these components are F y = p δx δy δz, y F z = p δx δy δz. z In total, therefore, the net pressure force is given by the vector F pressure = (F x,f y,f z ) µ p = x, p y, p z = p δxδyδz. δx δy δz (6.4) Note that the net force depends only on the gradient of pressure, p: clearly, a uniform pressure applied to all faces of the parcel would not introduce any net force.

9 6.2. EQUATION OF MOTION FOR A NON-ROTATING FLUID 161 Friction For typical atmospheric and oceanic flows, frictional effects are negligible except close to boundaries where the fluid rubs over the Earth s surface. The atmospheric boundary layer which is typically a few hundred meters to 1 km or so deep is exceedingly complicated. For one thing, the surface is not smooth: there are mountains, trees, and other irregularities that increase the exchange of momentum between the air and the ground. (This is the main reason why frictional effects are greater over land than over ocean.) For another, the boundary layer is usually turbulent, containing many smallscale and often vigorous eddies; these eddies can act somewhat like mobile molecules, and diffuse momentum more effectively than molecular viscosity. The same can be said of oceanic boundary layers which are subject, for example, to the stirring by eddies generated by the action of the wind, as will be discussed in Section At this stage, we will not attempt to describe such effects quantitatively but instead write the consequent frictional force on a fluid parcel as F fric = ρ F δx δy δz (6.5) where, for convenience, F is the frictional force per unit mass. Forthe moment we will not need a detailed theory of this term. Explicit forms for F will be discussed and employed in Sections and The equation of motion Putting all this together, Eq.(6.2) gives us ρδxδyδz Du = F gravity + F pressure + F fric, Substituting from Eqs.(6.3), (6.4), and (6.5), and rearranging slightly, we obtain Du + 1 p + gbz = F. (6.6) ρ This is our equation of motion for a fluid parcel. Note that because of our use of vector notation, Eq.(6.6) seems rather simple. However, when written out in component form, as below, it becomes somewhat intimidating, even in Cartesian coordinates:

10 162 CHAPTER 6. THE EQUATIONS OF FLUID MOTION u t + u u x + v u y + w u z + 1 p ρ x = F x (a) (6.7) v t + u v x + v v y + w v z + 1 p ρ y = F y (b) w t + u w x + v w y + w w z + 1 p ρ z + g = F z. (c) Fortunatelyoftenwewillbeabletomakeanumberofsimplifications. One such simplification, for example, is that, as discussed in Section 3.2, largescale flow in the atmosphere and ocean is almost always close to hydrostatic balance, allowing Eq.(6.7c) to be radically simplified as follows Hydrostatic balance From the vertical equation of motion, Eq.(6.7c), we can see that if friction and the vertical acceleration Dw/ are negligible, we obtain: p z = ρg (6.8) thus recovering the equation of hydrostatic balance, Eq.(3.3). For large-scale atmospheric and oceanic systems in which the vertical motions are weak, the hydrostatic equation is almost always accurate, though it may break down in vigorous systems of smaller horizontal scale such as convection Conservation of mass In addition to Newton s laws there is a further constraint on the fluid motion: conservation of mass. Consider a fixed fluid volume as illustrated in Fig.6.4. The volume has dimensions (δx, δy, δz). The mass of the fluid occupying this volume, ρδxδyδz, may change with time if ρ does so. However, mass 3 It might appear from Eq.(6.7c) that Dw/ << g is a sufficient condition for the neglect of the acceleration term. This indeed is almost always satisfied. However, for hydrostatic balance to hold to sufficient accuracy to be useful, the condition is actually Dw/ << g ρ/ρ, where ρ is a typical density variation on a pressure surface. Even in quite extreme conditions this more restrictive condition turns out to be very well satisfied.

11 6.3. CONSERVATION OF MASS 163 Figure 6.4: The mass of fluid contained in the fixed volume, ρδxδyδz, can be changed by fluxes of mass out of and in to the volume, as marked by the arrows. continuity tells us that this can only occur if there is a flux of mass into (or out of) the volume, i.e., t (ρδxδyδz)= ρ δx δy δz =(net mass flux into the volume). t Now the volume flux in the x-direction per unit time into the left face in Fig.6.4 is u x 1δx, y, z δy δz, so the corresponding mass flux is [ρu] x 1δx, y, z δy δz 2 2 where [ρu] is evaluated at the left face. That out through the right face is [ρu] x + 1δx, y, z δy δz; therefore the net mass import in the x-direction 2 intothevolumeis(againemployingataylorexpansion) (ρu) δx δy δz. x Similarly the rate of net import of mass in the y-direction is and in the z-direction is (ρv) δx δy δz y (ρw) δx δy δz. z

12 164 CHAPTER 6. THE EQUATIONS OF FLUID MOTION Therefore the net mass flux into the volume is (ρu) δx δy δz. Thusour equation of continuity becomes ρ + (ρu) =0. (6.9) t This has the general form of a physical conservation law: Concentration + (flux) =0 t in the absence of sources and sinks. Using the total derivative D/, Eq.(6.1), and noting that (ρu) =ρ u + u ρ (see the vector identities listed in Section 13.2) we may therefore rewrite Eq.(6.9) in the alternative, and often very useful, form: Dρ + ρ u =0. (6.10) Incompressible flow For incompressible flow (e.g. for a liquid such as water in our laboratory tank or in the ocean), the following simplified approximate form of the continuity equation almost always suffices: u = u x + v y + w =0; (6.11) z Indeed this is the definition of incompressible flow: it is non-divergent no bubbles allowed! Note that in any real fluid, Eq.(6.11) is never exactly obeyed. Moreover, despite Eq.(6.10), use of the incompressibility condition should not be understood as implying that Dρ =0; the density of a parcel of water can be changed by internal heating and/or conduction (see, for example, Section 11.1). While these density changes may be large enough to affect the buoyancy of the fluid parcel, they are too small to affect the mass budget. For example, the thermal expansion coefficient of water is typically K 1 and so the volume of a parcel of water changes by only 0.02% per degree of temperature change Compressible flow A compressible fluid such as air is nowhere close to being non-divergent ρ changes markedly as fluid parcels expand and contract. This is inconvenient

13 6.4. THERMODYNAMIC EQUATION 165 in the analysis of atmospheric dynamics. However it turns out that, provided the hydrostatic assumption is valid (as it nearly always is), one can get around this inconvenience by adopting pressure coordinates. In pressure coordinates, (x, y, p), the elemental fixed volume is δx δy δp. Since z = z (x, y, p), the vertical dimension of the elemental volume (in geometric coordinates) is δz = z δp and so its mass is δm given by: p δm = ρδxδyδz µ 1 p = ρ δx δy δp z = 1 g δx δy δp, where we have used hydrostatic balance, Eq.(3.3). So the mass of an elemental fixed volume in pressure coordinates cannot change! In effect, comparing the top and bottom line of the above, the equivalent of density in pressure coordinates the mass per unit volume is 1/g, a constant. Hence, in the pressure-coordinate version of the continuity equation, there is no term representingrateofchangeofdensity;itissimply p u p = u x + v y + ω p =0. (6.12) where the subscript p reminds us that we are in pressure coordinates. The greater simplicity of this form of the continuity equation, as compared to Eqs.(6.9) or (6.10), is one of the reasons why pressure coordinates are favored in meteorology. 6.4 Thermodynamic equation The equation governing the evolution of temperature can be derived from the first law of thermodynamics applied to a moving parcel of fluid. Dividing Eq.(4.12) by δt and letting δt 0 we find: DQ DQ = c DT p 1 Dp ρ. (6.13) is known as the diabatic heating rate per unit mass. In the atmosphere, this is mostly due to latent heating and cooling (from condensation and

14 166 CHAPTER 6. THE EQUATIONS OF FLUID MOTION evaporation of H 2 O) and radiative heating and cooling (due to absorption and emission of radiation). If the heating rate is zero then DT = 1 Dp : ρc p as discussed in Section 4.3.1, the temperature of a parcel will decrease in ascent (as it moves to lower pressure) and increase in descent (as it moves to higher pressure). Of course this is why we introduced potential temperature in Section 4.3.2: in adiabatic motion, θ is conserved. Written in terms of θ, Eq.(6.13) becomes µ κ Dθ p = Q. (6.14) p 0 c p where Q (with a dot over the top) is a shorthand for DQ. Here θ is given ³ p κ by Eq.(4.17), the factor p 0 converts from T to θ, and Q c p is the diabatic heating in units of Ks 1. The analogous equations that govern the evolution of temperature and salinity in the ocean will be discussed in Chapter Integration, boundary conditions and restrictions in application Eqs.(6.6), (6.11)/(6.12) and (6.14) are our five equations in five unknowns. Together with initial conditions and boundary conditions, they are sufficient to determine the evolution of the flow. Before going on, we make some remarks about restrictions in the application of our governing equations. The equations themselves apply very accurately to the detailed motion. In practice, however, variables are always averages over large volumes. We can only tentatively suppose that the equations are applicable to the average motion, such as the wind integrated over a 100 km square box. Indeed, the assumption that the equations do apply to average motion is often incorrect. The treatment of turbulent motions remains one of the major challenges in dynamical meteorology and oceanography. Finally, our governing equations have been derived relative to a fixed coordinate system. As we now go on to discuss, this is not really a restriction, but is usually inconvenient.

15 6.6. EQUATION OF MOTION FOR A ROTATING FLUID Equation of motion for a rotating fluid Eq.(6.6) is an accurate representation of Newton s laws applied to a fluid observed from a fixed, inertial, frame of reference. However, we live on a rotating planet and observe winds and currents in its rotating frame. For example the winds shown in Fig.5.20 are not the winds that would be observed by someone looking back at the earth, as in Fig.1. Rather, they are the winds measured by observers on the planet rotating with it. In most applications it is easier and more desirable to work with the governing equations in a frame rotating with the earth. Moreover it turns out that rotating fluids have rather unusual properties and these properties are often most easily appreciated in the rotating frame. To proceed, then, we must write down our governing equations in a rotating frame. However, before going on to a formal frame of reference transformation of the governing equations, we describe a laboratory experiment that vividly illustrates the influence of rotation on fluid motion and demonstrates the utility of viewing and thinking about fluid motion in a rotating frame GFD Lab III: Radial inflow We are all familiar with the swirl and gurgling sound of water flowing down a drain. Here we set up a laboratory illustration of this phenomenon and study it in rotating and non-rotating conditions. We rotate a cylinder about its vertical axis: the cylinder has a circular drain hole in the center of its bottom, as shown in Fig.6.5. Water enters at a constant rate through a diffuser on its outer wall and exits through the drain. In so doing, the angular momentum imparted to the fluid by the rotating cylinder is conserved as it flows inwards, and paper dots floated on the surface acquire the swirling motion seen in Fig.6.6 as the distance of the dots from the axis of rotation decreases. The swirling flow exhibits a number of important principles of rotating fluid dynamics conservation of angular momentum, geostrophic (and cyclostrophic) balance (see Section 7.1) all of which will be made use of in our subsequent discussions. The experiment also gives us an opportunity to think about frames of reference because it is viewed by a camera co-rotating with the cylinder.

16 168 CHAPTER 6. THE EQUATIONS OF FLUID MOTION Figure 6.5: The radial inflow apparatus. A diffuser of 30 cm inside diameter is placed in a larger tank and used to produce an axially symmetric, inward flow of water toward a drain hole at the center. Below the tank there is a large catch basin, partially filled with water and containing a submersible pump whose purpose is to return water to the diffuser in the upper tank. The whole apparatus is then placed on a turntable and rotated in an anticlockwise direction. The path of fluid parcels is tracked by dropping paper dots on the free surface. See Whitehead and Potter (1977).

17 6.6. EQUATION OF MOTION FOR A ROTATING FLUID 169 Figure 6.6: Trajectories of particles in the radial inflow experiment viewed in the rotating frame. The positions are plotted every 1 s. On the leftω =5rpm. On 30 the right Ω =10rpm. Note how the pitch of the particle trajectory increases as Ω increases and how, in both cases, the speed of the particles increases as the radius decreases. Observed flow patterns When the apparatus is not rotating, water flows radially inward from the diffuser to the drain in the middle. The free surface is observed to be rather flat. When the apparatus is rotated, however, the water acquires a swirling motion: fluid parcels spiral inwardascanbeseeninfig.6.6. Evenatmodest rotation rates of Ω =10rpm (corresponding to a rotation period of around 6 seconds) 4,theeffect of rotation is marked and parcels complete many circuits before finally exiting through the drain hole. The azimuthal speed of the particle increases as it spirals inwards, as indicated by the increase in the spacing of the particle positions in the figure. In the presence of rotation the free surface becomes markedly curved, high at the periphery and plunging downwards toward the hole in the center, as shown in the photograph, Fig An Ω of 10rpm (revolutions per minute) is equivalent to a rotation period τ = =6s. Various measures of table rotation rate are set out in Appendix

18 170 CHAPTER 6. THE EQUATIONS OF FLUID MOTION Figure 6.7: The free surface of the radial inflow experiment, in the case when the apparatus is rotated anticyclonically. The curved surface provides a pressure gradient force directed inwards that is balanced by an outward centrifugal force due to the anticlockwise circulation of the spiraling flow. Dynamical balances In the limit in which the tank is rotated rapidly, parcels of fluid circulate around many times before falling out through the drain hole (see the right hand frame of Fig.6.6); the pressure gradient force directed radially inwards (set up by the free surface tilt) is in large part balanced by a centrifugal force directed radially outwards. If V θ is the azimuthal velocity in the absolute frame (the frame of the laboratory) and v θ is the azimuthal speed relative to the tank (measured using the camera co-rotating with the apparatus) then (see Fig.6.8): V θ = v θ + Ωr (6.15) where Ω is the rate of rotation of the tank in radians per second. Note that Ωr is the azimuthal speed of a particle stationary relative to the tank at radius r fromtheaxisofrotation. We now consider the balance of forces in the vertical and radial directions, expressed firstintermsoftheabsolutevelocityv θ andthenintermsofthe relative velocity v θ.

19 6.6. EQUATION OF MOTION FOR A ROTATING FLUID 171 Figure 6.8: The velocity of a fluid parcel viewed in the rotating frame of reference: v rot =(v θ,v r ) in polar coordinates see Section Vertical force balance We suppose that hydrostatic balance pertains in the vertical, Eq.(3.3). Integrating in the vertical and noting that the pressure vanishes at the free surface (actually p = atmospheric pressure at the surface, whichherecanbetakenaszero),andwithρ and g assumed constant, we find that: p = ρg (H z) (6.16) where H(r) is the height of the free surface (where p =0)andwesuppose that z =0(increasing upwards) on the base of the tank (see Fig.6.5, left). Radial force balance in the non-rotating frame If the pitch of the spiral traced out by fluid particles is tight (i.e. in the limit that v r v θ << 1, appropriate when Ω is sufficiently large) then the centrifugal force directed radially outwards acting on a particle of fluid is balanced by the pressure gradient force directed inwards associated with the tilt of the free surface. This radial force balance can be written in the non-rotating frame thus: V 2 θ r = 1 p ρ r.

20 172 CHAPTER 6. THE EQUATIONS OF FLUID MOTION Using Eq.(6.16), the radial pressure gradient can be directly related to the gradient of free surface height enabling the force balance to be written 5 : V 2 θ r = g H r (6.17) Radial force balance in the rotating frame Using Eq.(6.15), we can express the centrifugal acceleration in Eq.(6.17) in terms of velocities in the rotating frame thus: Hence Vθ 2 r = (v θ + Ωr) 2 r v 2 θ = v2 θ r +2Ωv θ + Ω 2 r (6.18) r +2Ωv θ + Ω 2 r = g H (6.19) r Theabovecanbesimplified by measuring the height of the free surface relative to that of a reference parabolic surface 6, Ω2 r 2 as follows 2 Then, since η r = H Ω2 r r g η = H Ω2 r 2 2g. (6.20), Eq.(6.19) can be written in terms of η thus: vθ 2 r = g η r 2Ωv θ (6.21) Eq.(6.17) (non-rotating) and Eq.(6.21) (rotating) are completely equivalent statements of the balance of forces. The distinction between them is that the former is expressed in terms of V θ, the latter in terms of v θ. Note that Eq.(6.21) has the same form as Eq.(6.17) except (i) η (measured relative to the reference parabola) appears rather than H (measured relative to a flat surface) and (ii) an extra term, 2Ωv θ, appears on the rhs of Eq.(6.21) this is called the Coriolis acceleration. It has appeared because we have 5 Note that the balance Eq.(6.17) cannot hold exactly in our experiment because radial accelerations must be present associated with the flow of water inwards from the diffuser to the drain. But if these acceleration terms are small the balance (6.17) is a good approximation. 6 By doing so, we thus eliminate from Eq.(6.19) the centrifugal term Ω 2 r associated with the background rotation. We will follow a similar procedure for the spherical earth in Section (see also GFD Lab IV in Section 6.6.4).

21 6.6. EQUATION OF MOTION FOR A ROTATING FLUID 173 chosen to express our force balance in terms of relative, rather than absolute velocities. We shall see that the Coriolis acceleration plays a central role in the dynamics of the atmosphere and ocean. Angular momentum Fluid entering the tank at the outer wall will have angular momentum becausetheapparatusisrotating.atr 1, the radius of the diffuser in Fig.6.5, fluid has velocity Ωr 1 and hence angular momentum Ωr 2 1. As parcels of fluid flow inwards they will conserve this angular momentum (provided that they are not rubbing against the bottom or the side). Thus conservation of angular momentum implies that: V θ r = constant = Ωr 2 1 (6.22) where V θ is the azimuthal velocity at radius r in the laboratory (inertial) frame given by Eq.(6.15). Combining Eqs.(6.22) and (6.15) we find v θ = Ω (r2 1 r 2 ). (6.23) r We thus see that the fluid acquires a sense of rotation which is the same as that of the rotating table but which is greatly magnified at small r. IfΩ > 0 i.e. the table rotates in an anticlockwise sense then the fluid acquires an anticlockwise (cyclonic 7 )swirl. IfΩ < 0 the table rotates in a clockwise (anticyclonic) direction and the fluid acquires a clockwise (anticyclonic) swirl. This can be clearly seen in the trajectories plotted in Fig.6.6. Eq.(6.23) is, in fact, a rather good prediction for the azimuthal speed of the particles seen in Fig.6.6. We will return to this experiment later in Section where we discuss the balance of terms in Eq.(6.21) and its relationship to atmospheric flows Transformation into rotating coordinates In our radial inflow experiment we expressed the balance of forces in both the non-rotating and rotating frames. We have already written down the equations of motion of a fluid in a non-rotating frame, Eq.(6.6). Let us now 7 The term cyclonic (anticyclonic) means that the swirl is in the same (opposite) sense as the background rotation.

22 174 CHAPTER 6. THE EQUATIONS OF FLUID MOTION formally transform it in to a rotating reference frame. The only tricky part is the acceleration term Du/, which requires manipulations analogous to Eq.(6.18) but in a general framework. We need to figure out how to transform the operator D/ (acting on a vector) into a rotating frame. Of course, D/ of a scalar is the same in both frames, since this means the rate of change of the scalar following a fluid parcel. The same fluid parcel is followed from both frames, and so scalar quantities (e.g. temperature or pressure) do not change when viewed from the different frames. However, a vector is not invariant under such a transformation, since the coordinate directions relative to which the vector is expressed are different in the two frames. A clue is given by noting that the velocity in the absolute (inertial) frame u in and the velocity in the rotating frame u rot, are related (see Fig.6.9) through: u in = u rot + Ω r, (6.24) where r is the position vector of a parcel in the rotating frame, Ω is the rotation vector of the rotating frame of reference, and Ω r is the vector product of Ω and r. This is just a generalization (to vectors) of the transformation used in Eq.(6.15) to express the absolute velocity in terms of the relative velocity in our radial inflow experiment. As we shall now go on to show, Eq.(6.24) is a special case of a general rule for transforming the rate of change of vectors between frames, which we now derive. Consider Fig.6.9. In the rotating frame, any vector A may be written A =bxa x + bya y + bza z (6.25) where (A x,a y,a z ) are the components of A expressed instantaneously in terms of the three rotating coordinate directions, for which the unit vectors are (bx, by, bz). In the rotating frame, these coordinate directions are fixed, and so µ DA = bx DA x + by DA y rot + bz DA z However, viewed from the inertial frame, the coordinate directions in the.

23 6.6. EQUATION OF MOTION FOR A ROTATING FLUID 175 Figure 6.9: On the left is the velocity vector of a particle u in in the inertial frame. On the right is the view from the rotating frame. The particle has velocity u rot in the rotating frame. The relation between u in and u rot is u in = u rot + Ω r where Ω r is the velocity of a particle fixed (not moving) in the rotating frame at position vector r. The relationship between the rate of change of any vector A in the rotating frame and the change of A as seen in the inertial frame is given by: DA = DA in rot + Ω A.

24 176 CHAPTER 6. THE EQUATIONS OF FLUID MOTION rotating frame are not fixed, but are rotating at rate Ω, andso µ Dbx = Ω bx, in µ Dby = Ω by, in µ Dbz = Ω bz. in Therefore, operating on Eq.(6.25), µ DA = bx DA x + by DA y in + bzda z µ µ µ Dbx Dby Dbz + A x + A y + A z in in in = bx DA x + by DA y + bzda z +Ω (bxa x + bya y + bza z ), whence µ µ DA DA = + Ω A. (6.26) in rot which yields our transformation rule for the operator D acting on a vector. Setting A = r, the position vector of the particle in the rotating frame, we arrive at Eq.(6.24). To write down the rate of change of velocity following a parcel of fluidinarotatingframe, Du in, we set A u in in in Eq.(6.26) using Eq.(6.24) thus: µ µ Duin D = + Ω (u rot + Ω r) in rot µ Durot = +2Ω u rot + Ω Ω r, (6.27) rot since, by definition µ Dr = u rot. rot Eq.(6.27) is a more general statement of Eq.(6.18): we see that there is a one-to-one correspondence between the terms.

25 6.6. EQUATION OF MOTION FOR A ROTATING FLUID The rotating equation of motion We can now write down our equation of motion in the rotating frame. Substituting from Eq.(6.27) into the inertial-frame equation of motion (6.6), we have, in the rotating frame, Du + 1 p + gbz = 2Ω u ρ {z } Coriolis accel n + Ω {z Ω r } + F (6.28) Centrifugal accel n wherewehavedroppedthesubscripts rot and it is now understood that Du/ and u refer to the rotating frame. Note that Eq.(6.28) is the same as Eq.(6.6) except that u = u rot and apparent accelerations, introduced by the rotating reference frame, have been placed on the right-hand side of Eq.(6.28) [just as in Eq.(6.21)]. The apparent accelerations have been given names: the centrifugal acceleration ( Ω Ω r) is directed radially outward (Fig.6.9) the Coriolis acceleration ( 2Ω u) is directed to the right of the velocity vector if Ω is anticlockwise, sketched in Fig We now discuss these apparent accelerations in turn. Centrifugal acceleration As noted above, Ω Ω r is directed radially outwards. If no other forces were acting on a particle, the particle would accelerate outwards. Because centrifugal acceleration can be expressed as the gradient of a potential thus µ Ω 2 r 2 Ω Ω r = 2 where r is the distance ³ normal to the rotating axis (see Fig.6.9) it is convenient to combine 2 r 2 with gbz = (gz), the gradient of the gravitational Ω 2 potential gz, and write Eq.(6.28) in the succinct form: where Du + 1 p + φ = 2Ω u + F (6.29) ρ φ = gz Ω2 r 2 2 (6.30)

26 178 CHAPTER 6. THE EQUATIONS OF FLUID MOTION is a modified (by centrifugal accelerations) gravitational potential measured in the rotating frame. 8 In this way gravitational and centrifugal accelerations can be conveniently combined in to a measured gravity, φ. This is discussed in Section at some length in the context of experiments with a parabolic rotating surface GFD Lab IV. Coriolis acceleration The first term on the rhs of Eq.(6.28) is the Coriolis acceleration 9 it describes a tendency for fluid parcels to turn, as shown in Fig.6.10 and investigated in GFD Lab V. (Note that in this figure, the rotation is anticlockwise, i.e. Ω > 0, like that of the northern hemisphere viewed from above the north pole; for the southern hemisphere, the effective sign of rotation is reversed, see Fig.6.17.) In the absence of any other forces acting on it, a fluid parcel would accelerate as Du = 2Ω u (6.31) With the signs shown, the parcel would turn to the right in response to the Coriolis force (to the left in the southern hemisphere). Note that since, by definition, (Ω u) u = 0, the Coriolis force is workless: it does no work, but merely acts to change the flow direction. To breath some life in to these acceleration terms we will now describe experiments with a parabolic rotating surface. 8 Note that φ g =1 Ω2 r 2 2g is directly analogous to Eq.(6.20) adopted in the analysis of the radial inflow experiment. 9 Gustave Gaspard Coriolis ( ). French mathematician who discussed what we now refer to as the "Coriolis force" in addition to the already-known centrifugal force. The explanation of the effect sprang from problems of early 19th-century industry, i.e. rotating machines such as water-wheels.

27 6.6. EQUATION OF MOTION FOR A ROTATING FLUID 179 Figure 6.10: A fluid parcel moving with velocity u rot in a rotating frame experiences a Coriolis acceleration 2Ω u rot, directed to the right of u rot if, as here, Ω is directed upwards, corresponding to anticlockwise rotation GFD Lab IV and V: Experiments with Coriolis forces on a parabolic rotating table GFD Lab IV: studies of parabolic equipotential surfaces We fill a tank with water, set it turning and leave it until it comes in to solid body rotation, i.e., the state in which fluid parcels have zero velocity in the rotating frame of reference. This is easily determined by viewing a paper dot floating on the free surface from a co-rotating camera. We note that the free-surface of the water is not flat; it is depressed in the middle and rises up to its highest point along the rim of the tank, as sketched in Fig What s going on? In solid-body rotation, u = 0, F =0, and so Eq.(6.29) reduces to 1 ρ p + φ =0(a generalization of hydrostatic balance to the rotating frame). For this to be true: p ρ + φ = constant everywhere in the fluid (note that here we are assuming ρ = constant). Thus on surfaces where p =constant,φ must be constant too: i.e., p and φ surfaces must be coincident with one another. At the free surface of the fluid, p =0. Thus, from Eq.(6.30)

28 180 CHAPTER 6. THE EQUATIONS OF FLUID MOTION Figure 6.11: (a) Water placed in a rotating tank and insulated from external forces (both mechanical and thermodynamic) eventually comes in to solid body rotation in which the fluid does not move relative to the tank. In such a state the free surface of the water is not flat but takes on the shape of a parabola given by Eq.( 6.33). (b) parabolic free surface of water in a tank of 1 m square rotating at Ω =20rpm. gz Ω2 r 2 = constant (6.32) 2 the modified gravitational potential. We can determine the constant of proportionality by noting that at r =0,z= h(0), the height of the fluidinthe middle of the tank (see Fig.6.11a). Hence the depth of the fluid h is given by: h(r) =h(0) + Ω2 r 2 2g (6.33) where r is the distance from the axis or rotation. Thus the free surface takes on a parabolic shape: it tilts so that it is always perpendicular to the vector g (gravity modified by centrifugal forces) given by g = gbz Ω Ω r. If we hung a plumb line in the frame of the rotating table it would point in the direction of g i.e. slightly outwards rather than directly down. The surface given by Eq.(6.33) is the reference to which H is compared to define η in Eq.(6.20).

29 6.6. EQUATION OF MOTION FOR A ROTATING FLUID 181 Letusestimatethetiltofthefreesurfaceofthe fluid by inserting numbers into Eq.(6.33) typical of our tank. If the rotation rate is 10 rpm (so Ω ' 1 s 1 ) and the radius of the tank is 0.30 m, thenwithg =9.81 m s 2,wefind Ω 2 r 2 5mm,anoticeableeffect but a small fraction of the depth to which 2g thetankistypicallyfilled. If one uses a large tank at high rotation, however see Fig.6.11(b) in which a 1msquare tank was rotated at a rate of 20 rpm the distortion of the free surface can be very marked. In this case Ω2 r 2 2g 0.2m. It is very instructive to construct a smooth parabolic surface on which one can roll objects. This can be done by filling a large flat-bottomed pan withresinonaturntableandlettingtheresinhardenwhiletheturntable is left running for several hours (this is how parabolic surfaces are made). The resulting parabolic surface can then be polished to create a low friction surface. The surface defined by Eq.(6.32) is an equipotential surface of the rotating frame and so a body carefully placed on it at rest (in the rotating frame) should remain at rest. Indeed if we place a ball-bearing on the rotatingparabolicsurface andmakesurethatthetableisrotatingatthesame speed as was used to create the parabola! then we see that it does not fall in to the center, but instead finds a state of rest in which the component of gravitational force, g H resolved along the parabolic surface is exactly balanced by the outward-directed horizontal component of the centrifugal force, (Ω 2 r) H, as sketched in Fig.6.12 and seen in action in Fig GFD Lab V: visualizing the Coriolis force We can use the parabolic surfacediscussedinlabiv,inconjunctionwithadryice puck,tohelpus visualize the Coriolis force. On the surface of the parabola, φ = constant and so φ =0. We can also assume that there are no pressure gradients acting on the puck because the air is so thin. Furthermore, the gas sublimating off the bottom of the dry ice almost eliminates frictional coupling between the puck and the surface of the parabolic dish; thus we may also assume F =0. Hence the balance Eq.(6.31) applies. 10 We can play games with the puck and study its trajectory on the parabolic turntable, both in the rotating and laboratory frames. It is useful to view the puck from the rotating frame using an overhead co-rotating camera. Fig.6.14 plots the trajectory of the puck in the inertial (left) and in the rotating 10 A ball-bearing can also readily be used for demonstration purposes, but it is not quite as effective as a dry ice puck.

30 182 CHAPTER 6. THE EQUATIONS OF FLUID MOTION Figure 6.12: If a parabola of the form given by Eq.(6.33) is spun at rate Ω, then a ball carefully placed on it at rest does not fall in to the center but remains at rest: gravity resolved parallel to the surface, g H, is exactly balanced by centrifugal accelerations resolved parallel to the surface, (Ω 2 r) H. Figure 6.13: Studying the trajectories of ball bearings on a rotating parabola. A co-rotating camera views and records the scene from above.

31 6.6. EQUATION OF MOTION FOR A ROTATING FLUID 183 (right) frame. Notice that the puck is deflected to the right by the Coriolis force when viewed in the rotating frame if the table is turning anticlockwise (cyclonically). The following are useful reference experiments: 1. We place the puck so that it is motionless in the rotating frame of reference it follows a circular orbit around the center of the dish in the laboratory frame. 2. We launch the puck on a trajectory that crosses the rotation axis. Viewed from the laboratory the puck moves backwards and forwards alongastraightline(thestraightlineexpandsoutintoanellipseif thefrictionalcouplingbetweenthepuckandtherotatingdiscisnot negligible; see Fig.6.14a). When viewed in the rotating frame, however, the particle is continuously deflected to the right and its trajectory appears as a circle as seen in Fig.6.14b. This is the deflecting force of Coriolis. These circles are called inertial circles. (We will look at the theory of these circles below). 3. We place the puck on the parabolic surface again so that it appears stationary in the rotating frame, but is then slightly perturbed. In the rotating frame, the puck undergoes inertial oscillations consisting of small circular orbits passing through the initial position of the unperturbed puck. Inertial circles It is straightforward to analyze the motion of the puck in GFD Lab V. We adopt a Cartesian (x, y) coordinate in the rotating frame of reference whose origin is at the center of the parabolic surface. The velocity of the puck on the surface is u rot =(u, v) where u rot = dx/dt and v rot = dy/dt andwehavereintroducedthesubscript rot to make our frame of reference explicit. Further we assume that z increases upwards in the direction of Ω. Rotating frame Let us write out Eq.(6.31) in component form (replacing D by d, the rate of change of a property of the puck). Noting that (see dt VI, Appendix 13.2): 2Ω u rot =(0, 0, 2Ω) (u rot,v rot, 0) = ( 2Ωv rot, 2Ωu rot, 0) the two horizontal components of Eq.(6.31) are:

32 184 CHAPTER 6. THE EQUATIONS OF FLUID MOTION Figure 6.14: Trajectory of the puck on the rotating parabolic surface during one rotation period of the table 2π in (a) the inertial frame and (b) the rotating frame Ω of reference. The parabola is rotating in an anticlockwise (cyclonic) sense. du rot dt 2Ωv rot =0; dv rot dt +2Ωu rot =0 (6.34) u rot = dx dt ; v rot = dy dt. If we launch the puck from the origin of our coordinate system x(0) = 0; y(0) = 0 (chosen to be the center of the rotating dish) with speed u rot (0) = 0; v rot (0) = v o, the solution to Eq.(6.34) satisfying these boundary conditions is: u rot (t) =v o sin 2Ωt; v rot (t) =v o cos 2Ωt x (t) = v o 2Ω v o 2Ω cos 2Ωt; y (t) = v o sin 2Ωt (6.35) 2Ω The puck s trajectory in the rotating frame is a circle (see Fig.6.15) with a radius of vo. Itmovesaroundthecircleinaclockwisedirection(anticyclonically) with a period π, known as the inertial period. Note from Fig Ω Ω

33 6.6. EQUATION OF MOTION FOR A ROTATING FLUID 185 that in the rotating frame the puck is observed to complete two oscillation periodsinthetimeittakestocompletejustoneintheinertialframe. Inertial frame Now let us consider the same problem but in the nonrotating frame. The acceleration in a frame rotating at angular velocity Ω is related to the acceleration in an inertial frame of reference by Eq.(6.27). And so, if the balance of forces is Du rot = 2Ω u rot these two terms cancel out in Eq.(6.27), and it reduces to 11 : du in = Ω Ω r (6.36) dt If the origin of our inertial coordinate system lies at the center of our dish, then the above can be written out in component form thus: du in + Ω 2 x =0; dv in + Ω 2 y =0 (6.37) dt dt where the subscript in means inertial. This should be compared to the equation of motion in the rotating frame see Eq.(6.34). The solution of Eq.(6.37) satisfying our boundary conditions is: u in (t) =0; v in (t) =v o cos Ωt x in (t) =0; y in (t) = v o sin Ωt (6.38) Ω The trajectory in the inertial frame is a straight line shown in Fig Comparing Eqs.(6.35) and (6.38), we see that the length of the line marked out in the inertial frame is twice the diameter of the inertial circle in the rotating frame and the frequency of the oscillation is one-half that observed in the rotating frame, just as observed in Fig The above solutions go a long way to explaining what is observed in GFD Lab V and expose many of the curiosities of rotating versus non-rotating 11 Note that if there are no frictional forces between the puck and the parabolic surface, then the rotation of the surface is of no consequence to the trajectory of the puck. The puck just oscillates back and forth according to: d 2 r dh = g dt2 dr = Ω2 r wherewehaveusedtheresultthath, the shape of the parabolic surface, is given by Eq.(6.33). This is another (perhaps more physical) way of arriving at Eq.(6.37).

34 186 CHAPTER 6. THE EQUATIONS OF FLUID MOTION Figure 6.15: Theoretical trajectory of the puck during one complete rotation period of the table, 2π Ω, in GFD Lab V in the inertial frame (straight line) and in the rotating frame (circle). We launch the puck from the origin of our coordinate system x(0) = 0; y(0) = 0 (chosen to be the center of the rotating parabola) with v speed u(0) = 0; v(0) = v o. The horizontal axes are in units of o 2Ω. Observed trajectories are shown in Fig frames of reference. The deflection to the right by the Coriolis force is indeed a consequence of the rotation of the frame of reference: the trajectory in the inertial frame is a straight line! Before going on we note in passing that the theory of inertial circles discussed here is the same as that of the Foucault pendulum, named after the French experimentalist who in 1851 demonstrated the rotation of the earth by observing the deflection of a giant pendulum swinging inside the Pantheon in Paris. Observations of inertial circles Inertial circles are not just a quirk of this idealized laboratory experiment. They are in fact a common feature of oceanic flows. For example Fig.6.16 shows inertial motions observed by a current meter deployed in the main thermocline of the ocean at a depth of 500 m. The period of the oscillations is:

35 6.6. EQUATION OF MOTION FOR A ROTATING FLUID 187 Figure 6.16: Inertial circles observed by a current meter in the main thermocline of the Atlantic Ocean at a depth of 500 m; 28 N, 54 W. Five inertial periods are shown. The inertial period at this latitude is 25.6 h and 5 inertial periods are shown. Courtesy of Carl Wunsch, MIT. Inertial Period = π (6.39) Ω sin ϕ where ϕ is latitude; the sin ϕ factor (not present in the theory developed in Section 6.6.4) is a geometrical effect due to the sphericity of the Earth, as we now go on to discuss. At the latitude of the mooring, 28 N, the period of the inertial circles is 25.6 hrs Putting things on the sphere Hitherto our discussion of rotating dynamics has made use of a laboratory turntable in which Ω and g are parallel or antiparallel to one another. But on the spherical Earth Ω and g are not aligned and we must take into account these geometrical complications, illustrated in Fig We will now show that, by exploiting the thinness of the fluid shell (Fig.1.1) and the overwhelming importance of gravity, the equations of motion that govern the fluid on the rotating spherical Earth are essentially the same as those that govern the motion of the fluid of our rotating table if 2Ω is replaced by (what

36 188 CHAPTER 6. THE EQUATIONS OF FLUID MOTION Figure 6.17: In the rotating table used in the laboratory Ω and g are always parallel or (as sketched here) anti-parallel to one another. This should be contrasted with the sphere. is known as) the Coriolis parameter f =2Ωsinϕ. This is because it turns outthatafluid parcel on the rotating earth feels a rotation rate of only 2Ω sinϕ 2Ω resolved in the direction of gravity, rather than the full 2Ω. The centrifugal force, modified gravity and geopotential surfaces on the sphere Just as on our rotating table, so on the sphere the centrifugal term on the right of Eq.(6.28) modifies gravity and hence hydrostatic balance. For an inviscid fluid at rest in the rotating frame, we have 1 p = gbz Ω Ω r. ρ Now, consider Fig The centrifugal vector Ω Ω r has magnitude Ω 2 r,directedoutward normal to the rotation axis, where r =(a + z)cosϕ ' a cos ϕ, a is the mean Earth radius, z isthealtitudeabovethesphericalsurfacewithradiusa, and

37 6.6. EQUATION OF MOTION FOR A ROTATING FLUID 189 Figure 6.18: The centrifugal vector Ω Ω r has magnitude Ω 2 r, directed outward normal to the rotation axis. Gravity, g, points radially inwards to the center of the Earth. Over geological time the surface of the Earth adjusts to make itself an equipotential surface close to a reference ellipsoid which is always perpendicular the the vector sum of Ω Ω r and g. This vector sum is measured gravity: g = gbz Ω Ω r.

38 190 CHAPTER 6. THE EQUATIONS OF FLUID MOTION ϕ is latitude, and where the shallow atmosphere approximation has allowed us to write a + z ' a. Hence on the sphere Eq.(6.30) becomes: φ = gz Ω2 a 2 cos 2 ϕ 2 defining the modified gravitational potential on the Earth. At the axis of rotation the height of a geopotential surface is geometric height, z (because ϕ = π ). Elsewhere geopotential surfaces are defined by: 2 z = z + Ω2 a 2 cos 2 ϕ. (6.40) 2g We can see that Eq.(6.40) is exactly analogous to the form we derived in Eq.(6.33) for the free surface of a fluid in solid body rotation in our rotating tablewhenwerealizethatr = a cos ϕ is the distance normal to the axis of rotation. A plumb line is always perpendicular to z surfaces, and modified gravity is given by g = z. Since (with Ω = s 1 and a = m) Ω 2 a 2 /2g 11 km, geopotential surfaces depart only very slightly from a sphere, being 11 km higher at the equator than at the pole. Indeed, the figure of the Earth s surface thegeoid adoptssomethinglikethisshape,actuallybulging more than this at the equator (by 21 km, relative to the poles) 12. So, if we adopt these (very slightly) aspherical surfaces as our basic coordinate system, then relative to these coordinates the centrifugal force disappears (being subsumed into the coordinate system) and hydrostatic balance again is described (to a very good approximation) by Eq.(6.8). This is directly analogous, of course, to adopting the surface of our parabolic turntable as a coordinate reference system in GFD Lab V. Components of the Coriolis force on the sphere: the Coriolis parameter We noted in Chapter 1 that the thinness of the atmosphere allows us (for most purposes) to use a local Cartesian coordinate system, neglecting the Earth s curvature. First, however, we must figure out how to express the Coriolis force in such a system. Consider Fig The discrepancy between the actual shape of the Earth and the prediction Eq.(6.40) is due to the mass distribution of the equatorial bulge which is not taken in the calculation presented here.

39 6.6. EQUATION OF MOTION FOR A ROTATING FLUID 191 Figure 6.19: At latitude ϕ, longitude, λ, we define a local coordinate system such that the three coordinates in the (x, y, z) directions point (eastward, northward, upward): dx = a cos ϕdλ; dy = adϕ; dz = dz where a is the radius of the earth. The velocity is u =(u, v, w) in the directions (x, y, z). See also Section At latitude ϕ, we define a local coordinate system such that the three coordinates in the (x, y, z) directions point (eastward, northward, upward), as shown. The components of Ω in these coordinates are (0, Ω cos ϕ, Ω sin ϕ). Therefore, expressed in these coordinates, Ω u = (0, Ω cos ϕ, Ω sin ϕ) (u, v, w) = (Ω cos ϕw Ωsin ϕv,ωsin ϕu, Ωcos ϕu). We can now make two (good) approximations. First, we note that the vertical component competes with gravity and so is negligible if Ωu g. Typically, in the atmosphere u 10 m s 1,soΩu ms 2, which is utterly negligible compared with gravity (we will see in Chapter 9 that ocean currents are even weaker and so Ωu g there too). Second, because of the thinness of the atmosphere and ocean, vertical velocities (typically 1cm s 1 ) are very much less than horizontal velocities; so we may neglect the term involving w in the x-component of Ω u. Hence we may write the Coriolis term as 2Ω u ' ( 2Ω sin ϕv,2ωsin ϕu,0) (6.41) = fbz u,

40 192 CHAPTER 6. THE EQUATIONS OF FLUID MOTION latitude f ( 10 4 s 1 ) β ( s 1 m 1 ) Table 6.1: Values of the Coriolis parameter, f =2Ωsin ϕ Eq.(6.42) and its meridional gradient, β = df = 2Ω cos ϕ Eq.(10.10) tabulated as a function dy a of latitude. Here Ω is the rotation rate of the Earth and a is the radius of the Earth. where: f =2Ω sin ϕ (6.42) is known as the Coriolis parameter. NotethatΩ sin ϕ is the vertical component of the Earth s rotation rate this is the only component that matters (a consequence of the thinness of the atmosphere and ocean). For one thing this means that since f 0 at the equator, rotational effects are negligible there. Furthermore, f<0 in the southern hemisphere see Fig.6.17(right). Values of f at selected latitudes are set out in Table 6.1. We can now write Eq.(6.29) as (rearranging slightly), Du + 1 p + φ + fbz u = F. (6.43) ρ where 2Ω has been replaced by fbz. Writing this in component form for our local Cartesian system (see Fig.6.19), and making the hydrostatic approximation for the vertical component, we have Du + 1 p ρ x fv = F x, Dv + 1 p ρ y + fu = F y, (6.44) 1 p ρ z + g = 0,

41 6.6. EQUATION OF MOTION FOR A ROTATING FLUID 193 where (F x, F y ) are the (x, y) components of friction (and we have assumed the vertical component of F to be negligible compared with gravity). The set, Eq.(6.43) in component form, Eq.(6.44) is the starting point for discussions of the dynamics of a fluid in a thin spherical shell on a rotating sphere, such as the atmosphere and ocean GFD Lab VI: An experiment on the Earth s rotation A classic experiment on the Earth s rotation was carried out by Perrot in It is directly analogous to the radial inflow experiment, GFD Lab III, except that the Earth s spin is the source of rotation rather than a rotating table. Perrot filled a large cylinder with water (the cylinder had a hole in the middle of its base plugged with a cork, as sketched in Fig.6.20) and left it standing for two days. He returned and released the plug. As fluid flowed in toward the drain-hole it conserved angular momentum, thus concentrating the rotation of the Earth, and acquired a spin that was cyclonic (in the same sense of rotation as the Earth) According to theory below, we expect to see the fluid spiral in the same sense of rotation as the Earth. The close analogue with the radial inflow experiment is clear when one realizes that the container sketched in Fig 6.20 is on the rotating Earth and experiences a rotation rate of Ω sinlat! Theory We suppose that a particle of water initially on the outer rim of the cylinder at radius r 1 moves inwards conserving angular momentum until it reaches the drain hole, at radius r o see Fig The earth s rotation Ω earth resolved in the direction of the local vertical is Ω earth sin ϕ where ϕ is the latitude. Therefore a particle initially at rest relative to the cylinder at radius r 1, has a speed of v 1 = r 1 Ω earth sin ϕ in the inertial frame. Its angular momentum is A 1 = v 1 r 1.Atr o what is the rate of rotation of the particle? If angular momentum is conserved, then A o = Ω o ro 2 = A 1,andsotherate of rotation of the ball at the radius r o is: Ω o = µ r1 r o 2 Ω earth sin ϕ 13 Perrot s experiment can be regarded as the fluid-mechanical analogue of Foucault s 1851 experiment on the Earth s rotation using a pendulum.

42 194 CHAPTER 6. THE EQUATIONS OF FLUID MOTION Figure 6.20: A leveled cylinder is filled with water, covered by a lid and left standing for several days. Attached to the small hole at the center of the cylinder is a hose (also filled with water and stopped by a rubber bung) which hangs down in to a pail of water. On releasing the bung the water flows out and, according to theory, should acquire a spin which has the same sense as that of the Earth. ³ 2 r Figure 6.21: The Earth s rotation is magnified by the ratio 1 r o >> 1, ifthe drain hole has a radius r o, very much less than the tank itself, r 1.